scholarly journals Analytical assessment of the frequency of natural vibrations of a truss with an arbitrary number of panels

2020 ◽  
Vol 16 (5) ◽  
pp. 351-360
Author(s):  
Mikhail N. Kirsanov
Keyword(s):  
Energy Method ◽  
Oscillation Frequency ◽  
Frequency Equation ◽  
Linear Recurrence ◽  
Natural Oscillations ◽  
Analytical Estimate ◽  
Mass System ◽  
Symbolic Form ◽  
Upper Estimate ◽  
Homogeneous Linear

The aim of the work is to derive a formula for the dependence of the first frequency of the natural oscillations of a planar statically determinate beam truss with parallel belts on the number of panels, sizes and masses concentrated in the nodes of the lower truss belt. Truss has a triangular lattice with vertical racks. The solution uses Maple computer math system operators. Methods. The basis for the upper estimate of the desired oscillation frequency of a regular truss is the energy method. As a form of deflection of the truss taken deflection from the action of a uniformly distributed load. Only vertical mass movements are assumed. The amplitude values of the deflection of the truss is calculated by the Maxwell - Mohrs formula. The forces in the rods are determined in symbolic form by the method of cutting nodes. The dependence of the solution on the number of panels is obtained by an inductive generalization of a series of solutions for trusses with a successively increasing number of panels. For sequences of coefficients of the desired formula, fourth-order homogeneous linear recurrence equations are compiled and solved. Results. The solution is compared with the numerical one, obtained from the analysis of the entire spectrum of natural frequencies of oscillations of the mass system located at the nodes of the truss. The frequency equation is compiled and solved using Eigenvalue search operators in the Maple system. It is shown that the obtained analytical estimate differs from the numerical solution by a fraction of a percent. Moreover, with an increase in the number of panels, the error of the energy method decreases monotonically. A simpler lower bound for the oscillation frequency according to the Dunkerley method is presented. The accuracy of the lower estimate is much lower than the upper estimate, depending on the size and number of panels.

scholarly journals Natural Vibrations of the Reinforced Tapered Shell with Taking Into Account the Viscoelastic Properties of the Material

2021 ◽  
Vol 264 ◽  
pp. 01011
Author(s):  
Matlab Ishmamatov ◽  
Nurillo Kulmuratov ◽  
Nasriddin Ахmedov ◽  
Shaxob Хаlilov ◽  
Sherzod Ablakulov
Keyword(s):  
Conical Shell ◽  
Variational Principles ◽  
Variational Equation ◽  
Frequency Equation ◽  
Main Element ◽  
Algebraic Equations ◽  
Natural Vibrations ◽  
Natural Oscillations ◽  
Conical Shells ◽  
Truncated Conical Shell

In this paper, the integro-differential equations of natural oscillations of a viscoelastic ribbed truncated conical shell are obtained based on the Lagrange variational equation. The general research methodology is based on the variational principles of mechanics and variational methods. Geometrically nonlinear mathematical models of the deformation of ribbed conical shells are obtained, considering such factors as the discrete introduction of edges. Based on the finite element method, a method for solving and an algorithm for the equations of natural oscillations of a viscoelastic ribbed truncated conical shell with articulated and freely supported edges is developed. The problem is reduced to solving homogeneous algebraic equations with complex coefficients of large order. For a solution to exist, the main determinant of a system of algebraic equations must be zero. From this condition, we obtain a frequency equation with complex output parameters. The study of natural vibrations of viscoelastic panels of truncated conical shells is carried out, and some characteristic features are revealed. The complex roots of the frequency equation are determined by the Muller method. At each iteration of the Muller method, the Gauss method is used with the main element selection. As the number of edges increases, the real and imaginary parts of the eigenfrequencies increase, respectively.

The Study of Natural Oscillations of Thin Elastic Shells of Complicated Shapes by the Energy Method

2018 ◽  
Vol 931 ◽  
pp. 42-46
Author(s):  
Victor D. Eryomin
Keyword(s):  
Numerical Experiment ◽  
Energy Method ◽  
Natural Vibrations ◽  
Complicated Shape ◽  
Natural Oscillations ◽  
Elastic Shells ◽  
Numerical Convergence ◽  
Open Profile

The problem of natural oscillations of an elastic thin non-circular cylindrical wavy shell of an open profile is considered. The problem is based on the Rayleigh-Ritz energy method. On the basis of the proposed method for determining the lowest frequencies and forms of natural vibrations of shells of complicated shape, the numerical convergence of the developed algorithm is investigated. The evaluation of the results of this numerical experiment is given.

Experimental and theoretical investigation of large-amplitude oscillations of liquid droplets

1991 ◽  
Vol 231 ◽  
pp. 189-210 ◽  
Author(s):  
E. Becker ◽  
W. J. Hiller ◽  
T. A. Kowalewski
Keyword(s):  
Oscillation Frequency ◽  
Oscillation Amplitude ◽  
Nonlinear Effects ◽  
Finite Amplitude ◽  
Droplet Radius ◽  
Viscous Damping ◽  
Liquid Droplets ◽  
Axially Symmetric ◽  
Natural Oscillations ◽  
Gaseous Environment

Finite-amplitude, axially symmetric oscillations of small (0.2 mm) liquid droplets in a gaseous environment are studied, both experimentally and theoretically. When the amplitude of natural oscillations of the fundamental mode exceeds approximately 10% of the droplet radius, typical nonlinear effects like the dependence of the oscillation frequency on the amplitude, the asymmetry of the oscillation amplitude, and the interaction between modes are observed. As the amplitude decreases due to viscous damping, the oscillation frequency and the amplitude decay factor reach their asymptotical values predicted by linear theory. The initial behaviour of the droplet is described quite satisfactorily by a proposed nonlinear inviscid theoretical model.

scholarly journals Flexural vibrations of the walls of thin cylindrical shells having freely supported ends

1949 ◽  
Vol 197 (1049) ◽  
pp. 238-256 ◽  
Keyword(s):  
Experimental Investigation ◽  
Natural Frequency ◽  
Energy Method ◽  
Natural Frequencies ◽  
Frequency Equation ◽  
Diameter Ratio ◽  
Frequency Range ◽  
Thin Cylinder ◽  
Nodal Pattern ◽  
Modes Of Vibration

The paper deals with the general equations for the vibration of thin cylinders and a theoretical and experimental investigation is made of the type of vibration usually associated with bells. The cylinders are supported in such a manner that the ends remain circular without directional restraint being imposed. It is found that the complexity of the mode of vibration bears little relation to the natural frequency; for example, cylinders of very small thicknessdiameter ratio, with length about equal to or less than the diameter, may have many of their higher frequencies associated with the simpler modes of vibration. The frequency equation which is derived by the energy method is based on strain relations given by Timoshenko. In this approach, displacement equations are evolved which are comparable to those of Love and Flugge, though differences are evident due to the strain expressions used by each author. Results are given for cylinders of various lengths, each with the same thickness-diameter ratio, and also for a very thin cylinder in which the simpler modes of vibration occur in the higher frequency range. It is shown that there are three possible natural frequencies for a particular nodal pattern, two of these normally occurring beyond the aural range.

scholarly journals Lower estimate of the fundamental frequency of natural oscillations of a truss with an arbitrary number of panels

Vestnik MGSU ◽  
2019 ◽  
pp. 844-851
Author(s):  
Mikhail N. Kirsanov
Keyword(s):  
Fundamental Frequency ◽  
Arbitrary Number ◽  
Lower Estimate ◽  
Analytical Form ◽  
Equilibrium Equations ◽  
Rounding Errors ◽  
Natural Oscillations ◽  
Analytical Estimate ◽  
The Matrix ◽  
Analytical Result

Introduction: the paper deals with oscillations of a statically definable plane, truss with a double lattice of racks and descending braces with massive loads in the nodes of the lower chord. The weight of the truss rods is not taken into account. It is assumed that the freights are moved only vertically. The fundamental frequency of natural oscillations is estimated from the Dunkerley formula by the values of partial frequencies. Materials and methods: an analytical estimate is obtained by generalizing formulas obtained from a series of estimates for trusses with a consistently increasing number of panels. The stiffness of the truss was determined using the Mohr’s integral. The double lattice of the truss does not allow using the cross-section method; therefore, the forces in the rods were calculated (or estimated) in an analytical form using the method of cutting nodes with the compilation of a system of equilibrium equations simultaneously for all rods and three support reactions. The matrix of equilibrium equations was compiled in a software program written in the language of the Maple computer mathematics system based on the coordinates of the nodes and the values of the direction cosines of the forces. For a sequence of coefficients of the desired formula, linear homogeneous recurrent equations were found and solved by means of special operators of the Maple system. Results: the resulting formula estimating the relationship between the fundamental frequency and the panels number has the form of a sixth degree polynomial with coefficients depending on the parity of the number of panels. The analytical result is compared with the smallest frequency obtained numerically from the solution of the problem of oscillation of the cargo system. It is shown that the main frequency, depending on the truss height, has an extremum. Conclusions: the method of generalizing particular solutions using the Maple system operators allowed authors to obtain and analyze a formula for a lower estimate of the fundamental frequency of oscillation of a truss model with an arbitrary number of panels. The resulting estimate can be used as a test for numerically obtained solutions. The formula is especially efficient for systems with a large number of panels; as numerical methods for their calculation are time-consuming require considerable time and have a tendency for accumulating rounding errors.

scholarly journals Calculation of deformations of a cantilever-frame planar truss model with an arbitrary number of panels

Vestnik MGSU ◽  
2020 ◽  
pp. 510-517
Author(s):  
Karina Buka-Vaivade ◽  
Mikhail N. Kirsanov ◽  
Dmitrijs O. Serdjuks
Keyword(s):  
Analytical Solution ◽  
Numerical Solutions ◽  
Engineering Practice ◽  
Symbolic Form ◽  
Concentrated Load ◽  
Structural Rigidity ◽  
Truss Model ◽  
Different Types ◽  
Homogeneous Linear ◽  
Independent Parameters

Introduction. By method of induction using three independent parameters (numbers of panels) formulas for deflection under different types of loading are derived. Curves based on the derived formulas are analyzed, and the asymptotic of solutions for the number of panels are sought. The frame is statically definable, symmetrical, with descending braces. The problem of deflection under the action of a load evenly distributed over the nodes of the upper chord, a concentrated load in the middle of the span, and the problem of shifting the mobile support is considered. Materials and methods. The calculation of forces in the truss bars is performed in symbolic form using the method of cutting nodes and operators of the Maple computer mathematics system. The deflection is determined by the Maxwell – Mohr formula. Operators of the Maple computer mathematics system are used for composing and solving homogeneous linear recurrent equations that satisfy sequences of coefficients of the required dependencies. The stiffness of all truss bars is assumed to be the same. Results. All the obtained dependencies have a polynomial form for the number of panels. To illustrate the obtained solutions and their qualitative analysis, curves of the deflection dependence on the number of panels are constructed. Conclusions. A scheme of a statically definable three-parameter truss is proposed that allows an analytical solution of the problem of deflection and displacement of the support. The obtained dependences can be used in engineering practice in problems of structural rigidity optimization and for evaluating the accuracy of numerical solutions.

scholarly journals ANALYTICAL CALCULATION OF DEFLECTION OF A MULTI-LATTICE TRUSS WITH AN ARBITRARY NUMBER OF PANELS

2020 ◽  
Vol 16 (3) ◽  
pp. 23-29
Author(s):  
Mikhail Kirsanov
Keyword(s):  
Asymptotic Solution ◽  
Arbitrary Number ◽  
Analytical Calculation ◽  
Analytical Form ◽  
Linear Recurrence ◽  
Equilibrium Equations ◽  
Recurrence Equations ◽  
Cutting Out ◽  
Homogeneous Linear ◽  
System Of Equilibrium Equations

The scheme of a planar externally statically indeterminate truss with four supports is proposed. In analytical form, for several types of loads, the problem of forces in the rods and deflectionof the structure is solved, depending on the number of panels, the size and intensity of the load. The solution uses the Maple computer mathematics system. The deflectionat Midspan is determined using Maxwell – Mohr's formula, the forces in the rods – the method of cutting out nodes from the system of equilibrium equations for all nodes, which includes four reactions of the supports. By induction, a series of solutions for trusses with a consistently increasing number of panels is generalized to an arbitrary number of panels. For the elements of the sequences of coefficientare developed and are solved by homogeneous linear recurrence equations. The resulting formulas for the deflectio of the structure under various loads have the form of polynomials in the number of panels. A linear asymptotic solution for the number of panels is found. The kinematic degeneration of the structure and the distribution of node speeds corresponding to this case were found. The dependences of the reaction of supports and forces in the most compressed and stretched rods on the number of panels are determined.

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